On the Statistical Independence of Shift-Register Pseudorandom Multisequence over Part of the Period

TL;DR

This paper constructs a pseudorandom multisequence based on linear recurrences with low discrepancy over parts of its period, demonstrating strong statistical independence properties.

Contribution

It introduces a new construction of multisequences with provably low discrepancy and independence over segments, advancing pseudorandom sequence theory.

Findings

Discrepancy of the multisequence is bounded by O((N_1...N_r)^{-1/2} log^{s+3r}(N_1...N_r))

Sequence exhibits statistical independence over parts of its period

Construction based on kth-order linear recurrences modulo p

Abstract

In this paper we construct a pseudorandom multisequence $(x_{n_1,...,n_r})$ based on $k$th-order linear recurrences modulo $p$, such that the discrepancy of the $s$-dimensional multisequence $(x_{n_1+i_1,...,n_r+i_r})_{1 \leq i_j \leq s_j, 1 \leq j \leq r}$ $1 \leq n_j \leq N_j, 1 \leq j \leq r$ is equal to $O((N_1 ... N_r)^{-1/2} \ln^{s+3r}(N_1 ... N_r))$, where $s=s_1 ... s_r$, for all $N_1,...,N_r$ with $1 < N_1 ... N_r \leq p^k